Asymptotically Exact Relaxations for Robust LMI Problems based on Matrix-Valued Sum-of-Squares

In this paper we consider the problem of characterizing whether a symmetric polynomial matrix is positive definite on a semi-algebraic set. Based on suitable sum-of-squares representations we can construct LMI relaxation for this decision problem. As key novel technical contributions it is possible to prove that these relaxations are exact. Our proof is based on a sum-of-squares representation of r2 − ‖x‖2 with respect to affine functions with a priori constraints on the degree. This is a nontrivial extension of a rather deep result from [7] obtained by semidefinite duality arguments.